Advance successive linearization method to find critical eigenvalue in the rayleigh-bénard-brinkman convection problem with stress-free boundaries with isothermal conditions

An eigen boundary value problem is solved for the critical eigenvalue in the case of a Rayleigh-Bénard-Brinkman convection problem bounded by permeable walls with Robin boundary condition on temperature and general boundary condition on velocity. Unknown initial values and guess eigenvalue required to initiate the successive linearization method are obtained using the single-term Rayleigh-Ritz method. The local thermal non-equilibrium between the solid and fluid phases in the Newtonian fluid-saturated porous medium results in coupled partial differential equations. It is shown that five steps of the method results in a convergent solution with an accuracy of 6 decimal digits.